Deformation Spaces of Hyperbolic 3-manifolds

双曲3流形的变形空间

• 批准号: 0406976
• 负责人: Kenneth Bromberg
• 金额: $ 8.67万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406976&HistoricalAwards=false
• 关键词: Deformation; Spaces; Hyperbolic; manifolds

DMS-0204763Kenneth BrombergThe investigator plans to study spaces of hyperbolic metrics on a fixed3-manifold. Unlike on a closed manifold, on an open 3-manifold there willtypically be a large space of hyperbolic metrics. The investigatorsgoal is to understand this space. Some questions are simple to state:Is the deformation space the closure of its interior? A more detaileddescription of the space is the ending lamination conjecture,which gives invariants that classify every hyperbolic metric onthe manifold. This classification is not a parameterization; the map tothe space of invariants is not continuous. The investigator isinterested in understanding these discontinuities and describing thetopology of the deformation space of metrics. Hyperbolic cone-metricsare a key tool used throughout this work. The philosophy is that verycomplicated geometry in a smooth hyperbolic structure can be exchangedfor a less complicated, but singular, hyperbolic cone metric.Three manifolds are mathematical spaces that locally look like theuniverse we live in. Mathematicians, beginning with Poincare, have beeninterested in classification question about three manifolds. Hyperbolicgeometry is also a very old field dating back to the middle of theeighteenth century. In the last twenty-five years, largely due to the workThurston, it has been realized that there are deep and beautifulconnections between the two topics.

DMS-0204763 Kenneth Bromberg研究者计划研究固定三维流形上的双曲度量空间。与闭流形不同，在开的三维流形上通常会有一个很大的双曲度量空间。探索者的目标是了解这个空间。有些问题很容易表述：变形空间是其内部的封闭吗？一个更详细的描述的空间是结束层猜想，它给出了不变量，分类每个双曲度量流形上。这种分类不是一种参数化;到不变量空间的映射不是连续的。研究者对理解这些不连续性和描述度量变形空间的拓扑结构很感兴趣。双曲锥度量是贯穿这项工作的关键工具。其哲学思想是，光滑双曲结构中非常复杂的几何可以换成不那么复杂但奇异的双曲锥度量。三流形是局部看起来像我们生活的宇宙的数学空间。从庞加莱开始，数学家们一直对三维流形的分类问题感兴趣。双曲几何也是一个非常古老的领域，可以追溯到十八世纪中期。在过去的25年里，很大程度上由于瑟斯顿的工作，人们已经意识到这两个主题之间有着深刻而美丽的联系。